Topological Dynamics, Primorial Caging, and Non-Static Manifold Parameterisation in Discrete 3D Scaffolds

Abstract: This manuscript introduces a unified geometric and number-theoretic containment architecture designated as the Base-42 Primorial Cage Framework. Operating on a discrete 3D Conical Scaffold (\(\Pi_3 = 42\)) intersected by 12 Open Spoke Channels, we demonstrate a deterministic track collapse of the modular squaring operation \(x^2 \pmod{42}\) onto exactly three Master Rails (Rail 1, Rail 25, and Rail 37). This discrete foundation is continuously extended via Ford Circle gear trains to interface with spectral resonance eigenvalues (\(\lambda_1 = 0.0001400\)), a Collatz kinetic funnel macro-state, and the Riemann critical vacuum axis (\(\text{Re}(s) = 1/2\)). To validate the utility of this multi-scale caging architecture, we formalise eight structural asset containment classes: (A) a 13-channel RIS-13 manifold avionics telemetry stabiliser, (B) a lossless hypercomplex quaternionic vector nozzle pump constrained by \(\sigma^2 \le 1.083510 \times 10^{-9}\), (C) a non-associative 32D Trigintaduonion topological cage for fluid regularity boundaries, (D) a number-theoretic Sierpinski Coordinate Sieve isolating five unique multipliers (21181, 22699, 24737, 55459, 67607), (E) an Erdős–Straus fractional trajectory governor, (F) a geometric asymmetry framework for multiplicative divisor enclosures, (G) a Legendre density interval governor, and (H) an 18D phase-space gravitational chaos multi-body containment cell. Finally, we project these non-static physical implementation boundaries into pure dynamic time-varying projection maps \(\Psi: \mathcal{M} \rightarrow PG(4,2)\), establishing an anonymous, verifiable mathematical primitive for multi-domain systemic stabilization. Local Verification Fingerprint (SHA-256): 40082a2a4eee5ae3938abe700a2d13d981e8ff8bc2013d7f5b147eb9d784b07c